Existence and Exponential Decay for a Kirchhoff–Carrier Model with Viscosity

Abstract

In this work we study the existence of global solutions and exponential decay for the following nonlinear problem:

Existence and asymptotic behaviour for a degenerate kirchhoff-carrier model with viscosity and nonlinear boundary conditions

“…In order to obtain the existence of solutions in [3], the authors employ Galerkin's method and, as they are not allowed to use basis formed by eigenfunctions, make use of arguments which are only valid when the only nonlinearity is given by M = M (t). However, in the present paper , we have to deal with a nonlinear boundary condition and the degeneracy of the function M , which do not permit us to repeat the same arguments used in [3].…”

“…However, in the present paper , we have to deal with a nonlinear boundary condition and the degeneracy of the function M , which do not permit us to repeat the same arguments used in [3]. To overcome these difficulties is the goal of our work.…”

On a Nonlinear Wave Equation of Kirchhoff-Carrier Type: Linear Approximation and Asymptotic Expansion of Solution in a Small Parameter

“…Therefore, it is clear that (1) considered here contains (4) and (5) as special cases. Moreover, with various boundary conditions, the particular forms of (1) have been extensively studied by many authors; for example, we refer to [3][4][5][6][7][8][9][10][11][12][13][14][15] and the references given therein. In these works, many interesting results about existence, regularity, asymptotic behavior, asymptotic expansion, and decay of solutions were obtained.…”

“…Cavalcanti et al, in [4][5][6][7], investigated a series of four papers in which the results of existence, global existence, exponential or uniform decay rates, and asymptotic behavior for Kirchhoff-Carrier models are considered.…”

On Solvability of the Dissipative Kirchhoff Equation With Nonlinear Boundary Damping

“…For instance, we can see [18,19,21,22,23,30,31,34,35]. It is interesting to observe that problems with condition M (s) = 1 and a feedback occurs on the boundary were studied by many authors (see [3,4,5,6,7,12,13,14,20]). It is worth mentioning that M. Aassila [1] has employed Yosida approximation to obtain the global existence for the Kirchhoff equation with nonlinear damping as follows…”